Algebraic and Geometric Methods in StatisticsPaolo Gibilisco This up-to-date account of algebraic statistics and information geometry explores the emerging connections between the two disciplines, demonstrating how they can be used in design of experiments and how they benefit our understanding of statistical models, in particular, exponential models. This book presents a new way of approaching classical statistical problems and raises scientific questions that would never have been considered without the interaction of these two disciplines. Beginning with a brief introduction to each area, using simple illustrative examples, the book then proceeds with a collection of reviews and some new results written by leading researchers in their respective fields. Part III dwells in both classical and quantum information geometry, containing surveys of key results and new material. Finally, Part IV provides examples of the interplay between algebraic statistics and information geometry. Computer code and proofs are also available online, where key examples are developed in further detail. |
Contents
Indicator function and sudoku designs R Fontana and M | 12 |
Maximum likelihood estimation in latent class models for con | 27 |
Yuguo Chen | 99 |
Adrian Dobra | 135 |
interpolation and statistical modelling over | 159 |
Design of experiments and biochemical network inference | 175 |
Roberto Fontana | 203 |
Markov basis for design of experiments with threelevel factors | 225 |
Axiomatic geometries for text documents G Lebanon | 277 |
Exponential manifold by reproducing kernel Hilbert spaces | 291 |
Geometry of extended exponential models D Imparato | 307 |
Quantum statistics and measures of quantum information | 327 |
Algebraic varieties vs differentiable manifolds in statistical | 341 |
Coloured figures for Chapter 2 | 369 |
The generalised shuttle algorithm A Dobra and S E Fien | 395 |
Indicator function and sudoku designs R Fontana and M | 408 |
Introduction to nonparametric estimation R F Streater | 241 |
The Banach manifold of quantum states R F Streater | 257 |
Replicated measurements and algebraic statistics R Notari | 424 |
Common terms and phrases
algebraic geometry Algebraic Statistics algorithm analysis Annals of Statistics cell chapter characterisation compute connection consider constraints contingency tables convex coordinates corresponding defined Definition denote density dimension Dobra entropy equations equivalent example exponential family exponential model factorial design factors fiber field Fienberg find finite first Fisher information Fréchet bounds generalised Gibilisco given graphical models Grobner basis Hilbert hyperplane ideal identifiable independence indicator function information geometry integer interpolation intersection joint distribution kernel latent class models Lemma likelihood function linear log-likelihood log-linear model manifold Markov basis Markov chain Mathematics matrix maxima maximise methods metric monomial monomial ordering moves non-identifiable observed obtain odds ratios Orlicz space p-values parameter points polynomial probability problem Proof Proposition quantum quotient ring random variables Riccomagno Rogantin row and column sample satisfies Section simplex specification statistical models Sturmfels subset subspace sudoku suflicient statistic Theorem theory values vector Wynn zero